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Delta-Sigma Modulation the Art of Oversampling, Noise Shaping and Averaging

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Delta-Sigma Modulation the Art of Oversampling, Noise Shaping and Averaging lektronik abor Laboratory 2 Delta-Sigma Modulation The Art of Oversampling, Noise Shaping and Averaging Prof. Dr. Martin J. W. Schubert Electronics Laboratory Regensburg University of Applied Sciences Regensburg M. Schubert Lab2: Delta-Sigma Modulation Regensburg Univ. of Appl. Sciences Abstract. Quantization noise will be investigated in unmodulated and ΔΣ modulated form. Mathematical roots to understand the matter are discussed before practical evidence is given. Quantization noise power 2 2L+1 within the baseband 0...fB is proportional to Δ /OSR , with Δ being the quantizer’s minimum step, OSR=fs/2fB the oversampling ratio, fs the sampling frequency and L the modulator’s order. 1 Introduction (a) (b) u(t) x(n) sampled analog quantized sn Δ time time Fig. 1: (a) Analog and quantized signal, (b) sampled digital signal Any signal real is noisy. Mostly we try to reduce digital noise by a higher bit-width of the processed numbers. The ΔΣ modulator goes the opposite direction: It translates a long bit- vector or an analog quantity into a fast data stream with few parallel bits. The data stream encodes the input value within its average. This requires quantization [1]-[4]. Quantization is inevitable for analog-to-digital (A/D) conversion. Quantization and quantization noise correspond to the mathematical process of rounding and round-off noise. The minimum step of quantization is termed Δ. It can be a bit, a voltage, a current, etc. The power of noise shaping demonstrates Table 1. To improve the signal-to-noise ratio of the averaged signal by 60dB (= factor 1000 in voltage, 106 in power or ca. 8 bit signal width) an unmodulated signal must be OSR=106 times faster than required acc. to Nyquist. For a 1st order modulator an OSR of 126 is enough and for a 2nd order modulator an OSR of only 27! (PS: This calculation assumes that there are no other noise sources than quantization noise.) Table 1: OSR is the oversampling ratio and L the modulator’s order. Data was taken from [3]. SNRdB 20 dB 40 dB 60 dB 80 dB 100 dB OSR @ L=2 5 11 27 67 168 OSR @ L =1 6 28 126 578 2657 OSR @ L =0 100 10322 1,05E6 1,06E8 1,08E10 - 2 - M. Schubert Lab2: Delta-Sigma Modulation Regensburg Univ. of Appl. Sciences The theoretical part of ΔΣ modulation is relatively difficult [1]-[4]. But it is of high practical importance and a useful device to demonstrate numerous signal processing theorems. The practical part of this laboratory is relatively easy and many results are more or less given. Some documents help to understand equipment [5]-[8] and control loops [9]. The organization of this laboratory is as follows: Section 2 presents some very fundamental theoretical background on signal processing which most students should already know. Section 3 presents particular statements on ΔΣ modulation. Section 4 offers experimental verification. Section 5 draws relevant conclusion. - 3 - M. Schubert Lab2: Delta-Sigma Modulation Regensburg Univ. of Appl. Sciences 2 Some Fundamentals on Signal Processing 2.1 Correlation • Correlated signals depend on each other • Uncorrelated signals do not depend on each other Correlated signals sum in amplitude: ysum,corr = x1 + x2 + x3 + ...+ xN (2.1) 2 2 2 2 Uncorrelated signals sum in power: ysum,uncorr = x1 + x2 + x3 + ...+ xN (2.2) Different frequencies are always uncorrelated. (2.3) 2.2 Root-Mean-Square Computation of Effective Amplitudes 2.2.1 Effective Voltages and Currents and their RMS Computation Fig. 2.2.1: The effective DC voltage ueff{u(t)} dissipates the same power at a resistor R like the u(t). u(t)RR ueff The effective DC value of voltage u(t) is termed ueff{u(t)}. It dissipates the same power at a resistor R as the u(t). Good voltmeters either heat a reference resistor to the same temperature as u(t) or compute ueff by the Root Mean Square (rms) method: Square a signal, average it over time (mean) and then compute the square root of it. T 2 1 2 Time continuous: ueff = u = ∫ u (t)dt (2.4) T 0 N 2 1 2 Time discrete for samples xi, i=1...N: xeff = x = ∑ xi (2.5) N i=1 2 Note: The frequently observed expression u is 0 for an AC signal, because its mean value is 2 2 2 0 and 0 =0, too. Correct is averaging after squaring: ueff = u . 2.2.2 Effective Values of Some Particular Waveforms u (a) (b) u (c) rect usin tri Δ Δ Δ 2 2 2 0 00 t t t Δ Δ Δ 2 2 2 Fig. 2.2.2: Particular waveforms: (a) rectangular, (b) sinusoidal, (c) triangular. - 4 - M. Schubert Lab2: Delta-Sigma Modulation Regensburg Univ. of Appl. Sciences Fig. 2.2.2 shows (a) a rectangular, (b) a sinusoidal und (c) a triangular signal oscillating between the values Δ/2 and -Δ/2. Its total power at 1Ω and its effective voltage are given by 2 2 2 (Δ / 2) Δ Δ Δ / 2 Rectangular: urect = = ↔ urect,eff = = , (2.6) 1 4 2 1 2 2 2 (Δ / 2) Δ Δ Δ / 2 Sinusoidal: usin = = ↔ usin,eff = = , (2.7) 2 8 8 2 2 2 2 (Δ / 2) Δ Δ Δ / 2 Triangular: utri = = ↔ utri,eff = = . (2.8) 3 12 12 3 Adding an offset: The offset voltage is frequency 0Hz. Adding in power delivers 2 2 2 2 2 Combined: ucomb = uoffset + uxxx ↔ ucomb,eff = uoffset + uxxx,eff (2.9) with xxx standing for rect, sin, tri or others. 2.3 Estimating the Total Power of Quantization Noise Quantization is the process of expressing quantities as integral multiples of a minimum step, also termed Δ in signal processing. Quantization corresponds to the mathematical process of rounding and the quantization error eq corresponds to the mathematical round-off error. 2.3.1 Worst Case: Signal Power << Quantization Noise Power Δ averaged analog signal Fig. 2.3.1: Averaged analog signal u(t) is tiny 2 u(t) compared to the quantizer’s Δ/2. 0 t Δ 2 2-level quantized signal Fig. 2.3.1 illustrates the worst case scenario for a 2-level (1-bit) quantizer and a signal u(t) obtained from the bit-stream by averaging. This case is of very high practical relevance! Quantization noise is obtained from eq(t) = uideal(t) – uquantized(t) . (2.10) It is also clear that in the worst-case (2.10) becomes in the limit of u(t)→0 If |u(t)| << Δ/2 → eq(t) ≈ – uquantized(t) . (2.11) The total noise power and effective noise voltage is then given by (2.6): 2 u(t) / Δ →0 Δ u(t) / Δ →0 Δ Worst Case: |u(t)/Δ| → 0 : e2 ⎯⎯→⎯⎯⎯ ↔ e ⎯⎯→⎯⎯⎯ .(2.12) n,q,worst 4 n,q,worst,eff 2 - 5 - M. Schubert Lab2: Delta-Sigma Modulation Regensburg Univ. of Appl. Sciences 2.3.2 Best Case: Quantization Noise is Equidistributed Over Interval Δ (a) Fig. 2.3.2: yADC,out Illustrating the best-case quantization situation (a) ideal and real charac- teristics of a quan- UADC,in tizer real ideal (b) left : quanti- zation (b) eq eq noise eq . P (c) right: eq Δ equidis- UADC,in tributed over Δ. From signal processing literature [1]-[4] we know (without proof) that • If the input signal is sufficiently busy (no DC!) and (2.13) • the quantization error eq(t) is equidistributed over the interval Δ (2.14) then it can be shown [1]-[4], that the total noise power is described by (2.8), the effective value of a triangular oscillation: 2 2 Δ Δ Best Case: eq equidistributed over Δ : en,q,best = ↔ en,q,best,eff = . (2.15) 12 2 3 Δ2 Δ2 The reality will be somewhere between worst and best case, i.e. ≤ e2 ≤ . (2.16) 12 n,q 4 - 6 - M. Schubert Lab2: Delta-Sigma Modulation Regensburg Univ. of Appl. Sciences 2.4 Bel and Dezibel In honor of Graham Bell a factor 10 in signal power is termed a Bel, and 1 B = 10 dB, just as 1 m = 10 dm or 1 liter = 10 dl. As power corresponds to square of amplitude (p=u2/R=i2⋅R) and log(x2)=2⋅log(x) we get p2 p2 u2 i2 Signal-Ratio = log10 B =10log10 dB = 20log10 dB = 20log10 dB . (2.6) p1 p1 u1 i1 Question: 10dB is what factor in signal power? 10dB is what factor in effective voltage? 1 B = 10dB is a factor 10 in power → a factor 10 in voltage. 2.5 Estimating Signal Averages on Logarithmic Scale Printing averaged values on logarithmic axis delivers values in the upper range of the signal. • The average value of signal 2 sin2(πF) shown in Fig. 2.5(a) is 1 or 0dB. • The average value of signal 2 sin2(πF) 1/(1+πF) in Fig. 2.5(b) is 0.1061 or –19.5dB. Fig. 2.5: (a) Bode-plot of 2 sin2(πF) (b) Bode-plot of 2 sin2(πF) 1/(1+πF) 2.6 Time-Continuous versus Time-Discrete Models • Time-continuous systems are modeled in s (, i.e. by Laplace transformation). sT • Time-discrete systems are modeled in z=e with T=1/fs and fs sampling frequency. • In both cases – time-continuous and time-discrete – transfer functions are computed by setting s=jω. In signal processing we are rather interested in the relative frequencies F= f / fs and Ω = ω / fs than in the absolute frequencies f and ω=2πf.
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